Università Degli Studi Di Napoli Federico II

Transcription

1 Università Degli Studi Di Napoli Federico II Scuola Politecnica e delle Scienze di Base Dipartimento di Matematica e Applicazioni Renato Caccioppoli Tesi per il dottorato di ricerca in Scienze Matematiche e Informatiche XXIX Ciclo Optimization problems for nonlinear eigenvalues GIANPAOLO PISCITELLI

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3 C O N T E N T S introduction i preliminaries. Rearrangements. Convex symmetrization 5 anisotropic laplacian eigenvalue problems. Convex Symmetrization for Anisotropic Elliptic Equations with a lower order term.. Preliminary results.. Main result..3 Proof of main Theorem 4. On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators 9.. The Dirichlet eigenvalue problem for Q p 9.. The second Dirichlet eigenvalue of Q p 3..3 The limit case p 6.3 A sharp weighted anisotropic Poincaré inequality for convex domains Definition and statement of the problem Proof of the Payne-Weinberger inequality 34.4 The anisotropic -Laplacian eigenvalue problem with Neumann boundary conditions The limiting problem Proof of the Main Result Geometric properties of the first -eigenvalue 48 3 nonlocal problems A nonlocal anisotropic eigenvalue problem The first eigenvalue of the nonlocal problem On the First Twisted Dirichlet Eigenvalue The Nonlocal Problem 6 3. A saturation phenomenon for a nonlinear nonlocal eigenvalue problem Some properties of the first eigenvalue Changing-sign minimizers Proof of the main results 78

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5 I N T R O D U C T I O N This thesis is mainly focused on the study of some variational problems and elliptic partial differential equations that takes in to account a possible anisotropy. This kind of equations and functional arise from a generalization of the Euclidean case and are studied by means of symmetrization techniques, shape optimization and properties of Finsler metrics. On these questions Alvino, Bellettini, Ferone, Kawohl, Lions, Novaga, Trombetti (see e.g. [, 4,, ]) have obtained relevant results and later many authors have successifully continued the study of anisotropic eigenvalue problems (we refer for example to [44, 45, 46, 47, 48, 49, 5]). In the first Chapter we recall some definitions and properties of rearrangements referring to [6, 3, 78, 83, 94,, 7, 9]. We also introduce some notions of Finsler metrics, the definition of the Wulff shape and some generalized definitions and properties of perimeter, total variation, coarea formulas, isoperimetric inequalities [6, 4, 33, 3, ]. In the second chapter we study geometric properties of the eigenvalues of the anisotropic p-laplacian ( ) Q p u := div p ξ F p ( u), () with Dirichlet or Neumann boundary conditions, where F is a suitable norm (see Chapther for details) and < p +. In this chapter we study some isoperimetric problems, consisting in optimizing a domain dependent functional while keeping its volume fixed. Among the isoperimetric problems, we are interested in those ones linking the shape of domain to the sequence of its eigenvalue. Their study involves different fields of mathematics (spectral theory, partial differential equations, calculus of variations, shape optimization, rearrangement theory). One of the first question on optimization of eigenvalues appeared in the book of Lord Raylegh The theory of Sound (894). He conjectured that the first Dirichlet eigenvalue of Laplacian (the first frequency of the fixed membrane) is minimal for the disk. Thirty years later, Faber [6] and Krahn [86] proved this result with the means of rearrangements techniques, in particular by the Pólya-Szegö inequality []. Later Krahn [87], proved that for the second eigenvalue the minimizer is the union of two identic balls. Minimization of the first nontrivial Neumann eigenvalue of the Laplacian (the frequencies of the free membrane) among open sets of a given measure is a trivial problem, since the infimum is zero (even among convex sets). But it is bounded away from zero among convex sets with given diameter (Payne and Weinberger [97]). Moreover, contrary to the first Dirichlet eigenvalue, Szegö [6] (in the plane) and Weinberger [5] (in higher dimensions) proved that balls maximize the first notrivial Neumann eigenvalue among the open sets with given volume. One of the first attempt to solve a problem with an anisotropic function F, is contained in a paper of Wulff [6] dating back to 9. However, only in 944 A. Dinghas [54] proved that the set minimizing a generalized perimeter among open set with fixed volume is set homothetic to the unit ball of F o, the dual norm of F, i.e. W := {x R n : F o (x) }, () i

6 ii INTRODUCTION that is the so-called Wulff shape, centered in the origin. Moreover, we denote by W r (x ) the set rw + x, that is the Wulff shape centered in x with radius r and we put W r := W r (x ) if no misunderstanding occurs. This chapter is divided in four Section. In Section., by means of Schwarz (or spherical) symmetrization, it is possible to obtain comparison results for solutions to linear elliptic problems: where div(a(x, u, u)) = f in, u H () (3) a(x, η, ξ) ξ F (ξ) a.e. x, η R, ξ R n. (4) The authors in [4], using convex symmetrization, estimate a solution of (3) in terms of a function v that solves v = f # in #, v H ( # ), where f # is the spherically decreasing rearrangemetns of f (i.e. the function such that its level sets are balls which have the same measure as the level sets of f ) and # is the ball centered in the origin such that # =. In [99], we have considered a lower order term b(x, u) for (3), that is div(a(x, u, u)) + b(x, u) = f in, u H () (5) where a satisfies the ellipticity condition (4) and on b we assume that b(x, ξ) B(x)F(ξ) where B(x) is an integrable function. We have used convex symmetrization, to obtain comparison results with solutions of the convexly symmetric problem { div(f( v) Fξ ( v)) + b(f o (x))f( v)( F o ξ (x) F ξ( v)) = f in v H ( ), where F o is polar to F, b is an auxiliary function related to B, f is the convex rearrangement of f with respect to F (i.e. the function such that its level sets are Wulff shape which have the same measure as the level sets of f ) and is the set homothetic to the Wulff shape centered at the origin having the same measure as. We have obtained the following estimates: u v (6) F q ( u) F q ( v), (7) where < q and u is the convex rearrangement of u. The proof is based on some differential inequalities for rearranged functions obtained using Schwarz and Hardy inequalities and the properties of homogeneity and convexity of the function F. Finally we consider the case where b is essentially bounded by a constant β; we compare solutions of (5) with solutions to { div(f( v) Fξ ( v)) βf( v)( F o ξ (x) F ξ( v)) = f in v H ( )

7 INTRODUCTION iii and we obtain same estimates similar to (6) and (7) of the preceding case. In Section., our main aim is to study some properties of the Dirichlet eigenvalues of the anisotropic p-laplacian operator (). Namely, in [5], we analyze the values λ such that the problem { Qp u = λ(p, ) u p u in u = on, (8) admits a nontrivial solution in W,p (). Let us observe that the operator in () reduces to the p-laplacian when F is the Euclidean norm on R n and, for a general norm F, Q p is anisotropic and can be highly nonlinear. In literature, several papers are devoted to the study of the smallest eigenvalue of (8), denoted by λ (p, ), in bounded domains, which has the variational characterization λ (p, ) = min ϕ W,p ()\{} F p ( ϕ) dx. ϕ p dx Let be a bounded domain (i.e. an open connected set). It is known (see []) that λ (p, ) is simple, the eigenfunctions have constant sign and it is isolated and the only positive eigenfunctions are the first eigenfunctions. Furthermore, the Faber-Krahn inequality holds: λ (p, ) λ (p, ). Many other results are known for λ (p, ). The interested reader may refer, for example, to [, 5, 8, 48, 84]. As matter of fact, also different kind of boundary conditions have been considered as, for example, in the papers [44, 5] (Neumann case), [47] (Robin case). Among the results contained in the quoted papers, we recall that if is a bounded domain, it has been proved in [5] that lim λ (p, ) p = p ρ F (), where ρ F () is the radius of the bigger Wulff shape contained in, generalizing a well-known result in the Euclidean case contained in [8]. Actually, very few results are known for higher eigenvalues in the anisotropic case. In [69] the existence of a infinite sequence of eigenvalues is proved, obtained by means of a min max characterization. As in the Euclidean case, it is not known if this sequence exhausts all the set of the eigenvalues. Here we will show that the spectrum of Q p is a closed set, that the eigenfunctions are in C,α () and admit a finite number of nodal domains. We recall the reference [9], where many results for the spectrum of the p-laplacian in the Euclidean case have been summarized. The core of the result of this Section relies in the study of the second eigenvalue λ (p, ), p ], + [, in bounded open sets, defined as λ (p, ) := { min{λ > λ (p, ) : λ is an eigenvalue} if λ (p, ) is simple λ (p, ) and in analyzing its behavior when p. otherwise,

8 iv INTRODUCTION First of all, we show that if is a domain, then λ (p, ) admits exactly two nodal domains. Moreover, for a bounded open set, we prove a sharp lower bound for λ, namely the Hong-Krahn-Szego inequality λ (p, ) λ (p, W), where W is the union of two disjoint Wulff shapes, each one of measure. In the Euclidean case, such inequality is well-known for p =, and it has been recently studied for any < p < + in []. Finally, we address our attention to the behavior of λ (p, ) when is a bounded open set and p +. In particular, we show that lim λ (p, ) p = p ρ,f (), where ρ,f () is the radius of two disjoint Wulff shapes W, W such that W W is contained in. Furthermore, the normalized eigenfunctions of λ (p, ) converge to a function u that is a viscosity solution to the fully nonlinear elliptic problem: where A(u, u, u) = min{f( u) λu, Q u} = in, if u >, B(u, u, u) = max{ F( u) λu, Q u} = in, if u <, Q u = in, if u =, u = on, Q u = F ( u)( u ξ F( u)) ξ F( u). () In the Euclidean case, this kind of result has been proved for bounded domains in [8]. We consider both the nonconnected case and general norm F because our aim is twofold: first, to consider the case of a general Finsler norm F; second, to extend also the results known in the case of domains, to the case of nonconnected sets. In Section.3, we consider the set F (R n ) of lower semicontinuous functions, positive in R n \ {} and positively -homogeneous and we denote by L p ω() the weighted L p () space, where ω is a log concave function. In a general anisotropic case and for bounded convex domains of R n, we prove a sharp lower bound for the optimal constant Λ p,f,ω () in the Poincaré-type inequality inf u t t R Lω() p F( u) [Λ p,f,ω ()] p Lω (), p with < p < + and F F (R n ). If F is the Euclidean norm of R n and ω =, then Λ(p, ) = Λ p,e,ω () is the first nontrivial eigenvalue of the Neumann p Laplacian: { p u = Λ(p, ) u p u in, (9) u p u ν = on, then for a convex set, it holds that ( ) π p p Λ(p, ), diam E ()

9 INTRODUCTION v where + π p = + p (p ) p ds = π sp p sin π, diam E () Euclidean diameter of. p For other properties of π p and of generalized trigonometric functions, we refer to [9]. This estimate, proved in the case p = in [97] (see also [9]), has been generalized the ( case p ) = in [, 58, 66, ] and for p in [57, 4]. Moreover the constant π p p diam E () is the optimal constant of the one-dimensional Poincaré-Wirtinger inequality, with ω =, on a segment of length diam E (). When p = and ω =, in [7] an extension of the estimate in the class of suitable non-convex domains has been proved. Our aim, in [5], is to prove an analogous sharp lower bound for Λ p,f,ω (), in a general anisotropic case. More precisely, we prove the following inequality in a bounded convex domain R n, Λ p,f,ω () = inf u W, () u p uω dx= F( u) p ω dx u p ω dx ( π p diam F () ) p, () where diam F () = sup x,y F o (y x), < p < and ω is a positive log-concave function defined in. This result has been proved in the case p = and ω =, when F is a strongly convex, smooth norm of R n in [3] with a completely different method than the one presented here. In Section.4, we study the limiting problem of the anisotropic p-laplacian eigenvalue with Neumann boundary condition: { Qp u = Λ(p, ) u p u in ξ F p ( u) ν = on. () This problem is related to the Payne-Weinberger inequality (). In [], we study the the limit as p of eigenvalue problem (), we consider A(u, u, u) = min{f( u) Λu, Q u} = in, if u >, B(u, u, u) = max{ F( u) Λu, Q u} = in, if u <, Q u = in, if u =, F( u) ν = on, where ν is the outer normal to and Q is defined as in (). In the euclidean case (F( ) = ) this problem has been treated in [57, 4]. The solutions of (3) have be treated in viscosity sense and we refer to [3] and references therein for viscosity solutions theory and to [74] for Neumann problems condition in viscosity sense. Let us observe that for Λ = problem (3) has trivial solutions. We prove that all nontrivial eigenvalues Λ of (3) are greater or equal than: Λ(, ) := diam F (). This result has lots of interesting consequences. The first one is a Szegö-Weinberger inequality for convex sets, i.e. we prove that the Wulff shape, that has the same measure of, maximizes the first -eigenvalue among sets with prescribed measure: Λ(, ) Λ(, ). (3)

10 vi INTRODUCTION Then we prove that the first positive Neumann eigenvalue of (3) is never larger than the first Dirichlet eigenvalue of (9): Λ(, ) λ(, ), and that the equality holds if and only if is a Wulff shape. Finally we prove two important results regarding the geometric properties of the first nontrivial -eigenfunction. The first one shows that closed nodal domain cannot exist in ; the second one says that the first -eigenfunction attains its maximum only on the boundary of. In the third chapter we are interested in variational problems that are called nonlocal. This kind of problems are associated to non-standard Euler-Lagrange equations, in particular we consider equations perturbed with an integral term of the unknown function calculated on the entire domain. This kind of equations and functionals leads to a generalization of Sobolev inequality. On one hand, we introduce the non linearity with the means of a convex function F, on the other hand we add an integral term that represents the non-locality. Therefore, we study the optimal constant λ(α, ) in the following Sobolev-Poincaré inequality: [ ] ( u dx λ(α, ) ( ) ) (F( u)) dx + α u dx, u H(). In Section 3., we consider the following minimization problem with λ(α, ) = Q α (u, ) = inf Q α (u, ) (4) u H () (F( u)) dx + α( u dx) u dx where α is a real parameter. In this case, the Euler-Lagrange equation associated to problem (4) presents an integral term calculated over all, indeed the minimization problem (4) leads to the following eigenvalue problem { div(f( u) ξ F( u)) + α u dx = λu in, (5) u = on. In the euclidean case, when F(ξ) = ξ, problems like the above ones arise, for example, in the study of reaction-diffusion equations describing chemical processes (see [5]). More examples can be found in [9], [9], [43], [7] and [98]. The extension to a general F(ξ) is considered here as it has been made in other contexts to take into account a possible anisotropy of the problem. Typical examples are anisotropic elliptic equations ([4], []), anisotropic eigenvalue problems ([47], [48]), anisotropic motion by mean curvature ([], []). We also observe that, when α +, problem (4) becomes a twisted problem in the form (see [7] for the euclidean case) λ T () = inf u H () { F ( u) dx, u dx u dx = As in [7], we prove the following isoperimetric inequality λ T () λ T (W W ), }.

11 INTRODUCTION vii where W and W are two disjoint Wulff shape, each one with measure /. Now, our principal objective consists in finding an optimal domain which minimizes λ(α, ) among all bounded open sets with a given measure. If we denote with κ n the measure of W, in the local case (α = ) we have a Faber-Krahn type inequality λ(, ) λ(, ) = κ/n n j n/, /n, where j ν, is the first positive zero of J ν (z), the ordinary Bessel function of order ν, and is the Wulff shape centered at the origin with the same measure of. Hence, when α vanishes, the optimal domain is a Wulff shape. We show that the non local term affects the minimizer of problem (4) in the sense that, up to a critical value of α, the minimizer is again a Wulff shape, but, if α is big enough, the minimizer becomes the union of two disjoint Wulff shapes of equal radii. This is a consequence of the fact that the problem (4) have an unusual rescaling with respect to the domain. Indeed, we have λ(α, t) = t λ(tn+ α, ), which, for α =, becomes λ(, t) = λ(, ), t that is the rescaling in the local case. Therefore we show that we have a Faber-Krahn-type inequality only up to a critical value. Above this, we show a saturation phenomenon (see [7] for another example), that is the estimate cannot be improved and the optimal value remains constant. More precisely, in [] we prove, for every n, that, for every bounded, open set in R n and for every real number α, it holds { λ(α, ) if α +/n α c, where λ(α, ) α c = /n κn /n jn/, if α +/n α /n c, 3/n κ /n n j 3 n/, J n/,( /n j n/, ) /n j n/, J n/ ( /n j n/, ) nj n/ ( /n j n/, ). If equality sign holds when α +/n < α c then is a Wulff shape, while if inequality sign holds when α +/n > α c then is the union of two disjoint Wulff shapes of equal measure. In Figure we illustrate the transition between the two minimizers. λ /n j n, j n, O α c /κ +/n n α The continuous line represents the minimum of λ(α, ), among the open bounded sets of measure κ n, as a function of α.

12 viii INTRODUCTION In Section 3., we consider the following one-dimensional problem: { } λ(α, q) = inf Q[u, α], u H(, ), u, (6) where α R, q and Q[u, α] := u dx + α u dx u q u dx q. Let us observe that λ(α, q) is the optimal value in the inequality λ(α, q) u dx u dx + α q u q u dx. which holds for any u H (, ). Moreover, in the local case (α = ), this inequality reduces to the classical one-dimensional Poincaré inequality; in particular, λ(, q) = π 4 for any q. The minimization problem (6) leads, in general, to a nonlinear nonlocal eigenvalue problem. Indeed, supposing y y q dx, the associated Euler-Lagrange equation is ( ) y + α y y q q dx y q = λ(α, q) y in ], [ y( ) = y() =. Our purpose, in [53], is to study some properties of λ(α, q). In particular, depending on α and q, we aim to prove symmetry results for the minimizers of (6). Under this point of view, in the multidimensional case (N ) the problem has been settled out in [9] (when q = ) and in [43] (when q = ). We show that the nonlocal term affects the minimizer of problem (6) in the sense that it has constant sign up to a critical value of α and, for α larger than the critical value, it has to change sign, and a saturation effect occurs. For q, we prove that there exists a positive number α q such that, if α < α q, then λ(α, q) < π, and any minimizer y of λ(α, q) has constant sign in ], [. If α α q, then λ(α, q) = π. Moreover, if α > α q, the function y(x) = sin πx, x [, ], is the only minimizer, up to a multiplicative constant, of λ(α, q). Hence it is odd, y(x) q y(x) dx =, and x = is the only point in ], [ such that y(x) =. Furthermore, we analyze the behaviour of the minimizers for the critical value α = α q. If q =, we have α = π. Moreover, if α = α, there exists a positive minimizer of λ(α, ), and for any x ], [ there exists a minimizer y of λ(α, ) which changes sign in x, non-symmetric and with y(x) dx = when x =. If < q and α = α q, then λ(α q, q) in [, ] admits both a positive minimizer and the minimizer y(x) = sin πx, up to a multiplicative constant. Hence, any minimizer has constant sign or it is odd.

13 INTRODUCTION ix Let us observe that, for any α R, it holds that λ(α, q) Λ q = π, where Λ q := min u dx, u H(, ), u dx u q u dx =, u. (7) It is known that, when q [, ], then Λ q = Λ = π, and the minimizer of (7) is, up to a multiplicative constant, y(x) = sin πx, x [, ] (see for example [34]). Problems with prescribed averages of u and boundary value conditions have been studied in several papers. We refer the reader, for example, to [3, 7, 34, 55, 56, 75, 95]. In recent literature, also the multidimensional case has been adressed (see, for example [8, 7, 35, 36, 96]). Finally, I wish to express my deep gratitude to my supervisor, Professor Vincenzo Ferone, for his valuable teaching during my three years work under his guidance. I am grateful to Cristina Trombetti, Carlo Nitsch, Francesco Della Pietra and Nunzia Gavitone for their helpful suggestions during the preparation of the present thesis. I gratefully acknowledge Professor Bernd Kawohl for all the useful scientific advices and for the support that he gave me at the time of my stay in Cologne.

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15 P R E L I M I N A R I E S. rearrangements Let be a measurable and not negligible subset of R n. We denote with its n- dimensional Lebesgue measure. Let R n n the space of real matrices, we denote the matrix product between two matrices A, B R n n by (AB). Let ξ, ζ R n, we denote the scalar product between ξ and ζ by ξ ζ. We recall some definitions and properties of rearrangements (we refer to [3, 65, 85, 83]). Definition.. Let u : R be a measurable function. We define the distribution function of u as the map µ : [, [ [, [ such that µ(t) := {x : u(x) > t}. (8) Such function represents the measure of the level sets of u and satisfies the following properties. Proposition.. Let µ defined as in (8), then. µ( ) is monotone decreasing;. µ() = supp u ; 3. supp µ = [, ess sup u ]; 4. µ( ) is right-continuous; 5. µ(t ) µ(t) = {x : u(x) = t}. Proof. Properties (), () and (3) follows immediately from the definition. To prove (4) and (5), let us observe that {x : u(x) > t} = k= { x : u(x) > t + } k and {x : u(x) > t} = k= { x : u(x) > t }. k Therefore { µ(t + ) = lim x : u(x) > t + } = µ(t), k k and { µ(t ) = lim x : u(x) > t + } = µ(t) + {x : u(x) = t}. k k

16 PRELIMINARIES We stress that the distribution function µ is discontinuous only for the value t such that {x : u(x) = t} =. Definition.3. Let u : R be a measurable function. We define the decreasing rearrangement of u as the map u : [, [ [, [ such that u (s) = sup{t > : µ(t) > s}. The decreasing rearrangements is a generalization of the inverse of µ in the sense that u is the distribution function of µ. Therefore u satisfies the following:. u is monotone decreasing;. u is right-continuous; 3. u () = ess sup u; 4. supp u = [, supp u ]; 5. u (µ(t)) t and µ(u (s)) s. Definition.4. Let u and v : R two measurable function, we say that u and v are equimisurable if they have the same distribution function. Proposition.5. The functions u : R and u : [, ] [, [ are equimisurable, that is for all t, {x : u(x) > t} = {s [, ] : u (s) > t}. (9) Proof. By the definition of u, it follows that Hence we have if u (s) > t, then s < µ(t); if u (s) t, then s µ(t). {s : u (s) > t} = [, µ(t)[, that gives (9). Proposition.6. Let u L p (, R) and p, then u L p (, ), and u L p () = u L p (, ). () Proof. If p <, then, by Cavalieri s principle we have u p dx = + µ(t)d(t p ). Hence () follows from Proposition.5. If p =, the result follows from the definition of rearrangement. Corollary.7. Let u : R be a measurable function, we have {x : u(x) t} = {s [, ] : u (s) t}. () {x : u(x) < t} = {s [, ] : u (s) < t}. () {x : u(x) t} = {s [, ] : u (s) t}. (3)

17 REARRANGEMENTS 3 Proof. Since (3) is equivalent to (9) and () is equivalent to (), then it is sufficient to prove that (9) and () are equivalent. Indeed, being and lim {x : u(x) > t + h} = {x : u(x) > t} h + lim {x : u(x) > t h} = {x : u(x) t}, h + we get the thesis. Definition.8. We denote by # the ball centered in the origin having the same measure as. Let u : R, we define the sferically decreasing rearrangement or Schwarz symmetrization of u, as the map u # : # [, [ such that u # (x) = u (ω n x n ), x #, where ω n is the measure of the unit ball in R n, namely ω n = π n Γ ( n + ). For other details we refer to [83]. Now we state the following Theorem, which says that a function in W,p () is also in W,p ( # ) and that the L p -norm of the gradient decreases under the effect of rearrangement. Theorem.9. If u W,p (R n ), p < +, is a nonnegative function with compact support, then u # W,p (R n ) and u # p dx u p dx. R n R n To introduce the notion of perimeter given by De Giorgi in [4], we define the bounded variation function. For these results we mainly refer to [63, 6, 7, 7]. We denote by B() the σ-algebra of all Borel subsets of. Definition.. Let ν : B() R k a vector-valued Radon measure on, we define the total variation of ν in by { } k ν () = sup ϕ(x) dν i (x) : ϕ C (, R k ), ϕ. i= Definition.. Let u L (), we say that u is a function with bounded variation in, shortly a BV function, if there exists a Radon measure λ with values in R n such that for any i =,..., n and any ϕ C () u ϕ dx = ϕ dλ i. x i The measure λ is also called the measure derivative of u and is denoted by the symbol Du. By BV() we denote the vector space of functions with bounded variations in. This space can be endowed with the norm u BV() = u L () + D u (), thus becoming a Banach space. We mainly use the following characterization of the BV functions.

18 4 PRELIMINARIES Theorem.. Let u L (). Then u BV() if and only if { } V = sup u(x) div ϕ(x) dx : ϕ C(, R n ), ϕ <. (4) Moreover the supremum V in (4) is equal to the total variation Du () of Du in. We recall that a sequence ν h of Radon measures is said to converge weakly* to a Radon measure ν if lim h i= k ϕ i d(ν h ) i = i= k ϕ i dν i for any ϕ C c (, R k ). Moreover a sequence u h in BV() converges weakly* to a BV() function u if u h u in L () and Du h converges to Du weakly* in in the sense of measures. The following approximation theorem states that we can approximate the BV functions by smooth functions in the weak* convergence sense. Theorem.3. Let u BV(). Then, there exists a sequence {u h } h N C () BV() such that u h u weakly* in BV(), lim u h dx = Du () h Proof. We give the result in the case = R n. The general case is proved using the same methods, only with some extra technicalities. Let ρ be a positive, radially symmetric function with compact support in B, such that R n ρ dx =. For all ε >, we set ρ ε (x) = ε n ρ( x ε ) and u ε (x) = (u ρ ε )(x) = ρ ε (x y)u(y) dy. R n By the properties of mollified functions, we have that u ε u in L (R n ). Moreover, for any x R n u ε (x) = u(y) ρ ε (x y) dy = u(y) ρ ε (x y) dy = ρ ε (x y) dd i u(y), x i R n x i R n y i R n for i =,..., n. Now we fix ϕ C (R n ), with ϕ and, from (5), we get n ϕ u ε dx = ϕ i (x) dx ρ ε (x y)dd i u(y) R n i= R n R n n n = dd i u(y) ρ ε (x y)ϕ i (x) dx = (ϕ i ρ ε )(y) dd i u(y) i= R n R n i= R n Since also ϕ ρ ε, we can take the supremum over all such functions ϕ to obtain R n u ε dx Du (R n ). (6) Therefore the measures u ε dx converge weakly* to the measure Du and thus, by the lower semicontinuity of the total variation, we have Du (R n ) lim inf u ε dx. ε R n This inequality, together with (6), concludes the proof. Now we give the definition of perimeter. (5)

19 CONVEX SYMMETRIZATION 5 Definition.4. Let E be a measurable subset of R n and an open set. The perimeter of E in is defined by the quantity { } P(E; ) = sup div ϕ dx : ϕ C(; R n ), ϕ. E If P(E; ) <, we say that E is a set of finite perimeter in. We recall that by the use of Theorem.3 we have the following approximation result for sets of finite perimeter. We shall write simply P(E) to denote the perimeter of E in R n. Theorem.5. Let E be a set of finite perimeter in R n with E <. Then, there exists a sequence of bounded open sets E h with C boundaries, such that χ Eh χ E in L (R n ) and P(E h ) P(E). Finally we state the isoperimetric inequality (we refer to [73, 8]). Theorem.6. Let E R n be a set of finite perimeter with finite measure. Then, P(E) nω n n E n. Proof. By Theorem.3 we can approximate χ E by a sequence of function such that u h (x) χ E (x) a.e x R n, R n u h dx Dχ E (R n ) = P(E). Then follows immediately by the classical Sobolev imbedding Theorem ( u h dx nω n u h n n R n R n dx ) n n.. convex symmetrization Throughout this thesis we will consider a convex even -homogeneous function (see also [, 4, 33, 3]) ξ R n F(ξ) [, + [, that is a convex function such that F(tξ) = t F(ξ), t R, ξ R n, (7) and such that a ξ F(ξ), ξ R n, (8) for some constant < a. Under this hypothesis it is easy to see that there exists b a such that F(ξ) b ξ, ξ R n.

20 6 PRELIMINARIES Moreover, we assume that ξ [Fp ](ξ) is positive definite in R n \ {}, (9) with < p < +. The hypothesis (9) on F assures that the operator ( ) Q p [u] := div p ξ[f p ]( u) is elliptic, hence there exists a positive constant γ such that n p ξ i ξ j [F p ](η)ξ i ξ j γ η p ξ, i,j= for some positive constant γ, for any η R n \ {} and for any ξ R n. Remark.7. We stress that for p the condition ξ [F ](ξ) is positive definite in R n \ {}, implies (9). The polar function F o : R n [, + [ of F is defined as F o (v) = sup ξ = ξ v F(ξ). It is easy to verify that also F o is a convex function which satisfies properties (7) and (8). Furthermore, F(v) = sup ξ = ξ v F o (ξ). From the above property it holds that The set ξ η F(ξ)F o (η), ξ, η R n. (3) W = {ξ R n : F o (ξ) < } is the so-called Wulff shape centered at the origin. We put κ n = W, where W denotes the Lebesgue measure of W. More generally, we denote with W r (x ) the set rw + x, that is the Wulff shape centered at x with measure κ n r n, and W r () = W r. The following properties of F and F o hold true (see for example []): ξ F(ξ) ξ = F(ξ), ξ F o (ξ) ξ = F o (ξ), (3) F( ξ F o (ξ)) = F o ( ξ F(ξ)) =, ξ R n \ {}, (3) F o (ξ) ξ F( ξ F o (ξ)) = F(ξ) ξ F o ( ξ F(ξ)) = ξ ξ R n \ {}, (33) n ξ i ξ j F(ξ)ξ j =, i =,..., n. (34) j=

21 CONVEX SYMMETRIZATION 7 Definition.8. Let u BV(), we define the total variation of u with respect to F as { } Du F = sup u div σ dx : σ C(; R n ), F o (σ) and the perimeter of a set E with respect to F: P F (E; ) = { } Dχ E F = sup div σ dx : σ C(; R n ), F o (σ) E These definition yields to the following co-area formula Du F = and to the equality P F (E; ) = E P F ({u > s}; ) ds F(ν E ) dh n (x), u BV() where E is the reduced boundary of E and ν E is the outer normal to E (see also [6]). Definition.9. We denote by the Wulff shape centered in the origin having the same measure as. Let u : R, we define the (decreasing) convex rearrangement of u (see [4]) as the map u : [, [, such that u (x) = u (κ n (F o (x)) n ). (35) By definition it holds u L p () = u L p ( ), for p +. Furthermore, when u coincides with its convex rearrangement, we have (see [4]) u (x) = u (κ n (F o (x)) n )nκ n (F o (x)) n ξ F o (x); (36) F( u (x)) = u (κ n (F o (x)) n )nκ n (F o (x)) n ; (37) ξ F( u (x)) = x F o (x). (38) Now, we recall here a result about a Pólya-Szegö principle related to H (we refer to [4], []) in the equality case (see [59], [64] for further details). Proposition.. Let u W,p (), p. Then u W,p () (F( u)) dx (F( u )) ds. (39) Furthermore, if u satisfies the equality in (39), then, for a.e. t [, ess sup u], the set {x : u(x) > t} is equivalent to a Wulff shape. Proposition.. Let E be a subset of R n. Then P F (E; ) is finite if and only if the usual perimeter P(E; ) is finite. Moreover we have αp(e; ) P F (E; ) βp(e; ).

22 8 PRELIMINARIES Proof. By (7) and (8), we have β ξ Fo (ξ) α ξ, ξ Rn. and hence the result follows. To show an isoperimetric inequality which estimate from below the perimeter with respect to a gauge function F of a set E, we give the following approximation results. Proposition.. Let u BV(). A sequence {u h } h N C () exists, such that: and lim u h u = h lim Du h F = Du F. h Proof. By mollifying u, we define a sequence {u h } h N with the required properties following the proof of Theorem.7 of [76]. Proposition.3. Let E be a set of finite perimeter in. There exists a sequence {E h } h N of C sets such that: χ Eh χ E = and lim h lim Dχ Eh F = P F (E; ) h Proof. We find the proof in [4]. We mollify the function χ E as in Proposition., hence we find a sequence { f h } h N C () such that: and lim f h χ E = h lim D f h F = P F (E; ). h By the coarea formula we have u h F = P F ({u h > s}; )ds. Sard s theorem implies that the sets E (h) s = {u h > s} have C boundary for almost every s (, ) and We consider only such levels s. Let us fix ε ], 4 [ and h = h(ε) such that: f h χ E < ε. Arguing as in [93, Lemma, p.99], we get χ E χ (h) E < ε, (4) s

23 CONVEX SYMMETRIZATION 9 for every s [ε, ε ]. On the other hand, for every h there exists s h (ε, ε ) such that: ( ε )PF (E (h) s h ; ) Moreover we have P F (E; ) = lim h P F (E (h) t ; ) dt. (4) D f h F = lim h P F (E (h) t ; ) dt. (4) By (4), it follows that χ E (h) s h χ E in L () and by (4) and (4) we have lim sup ε P F (E (h) s h ; ) P F (E; ). Since P F is lower semicontinuous, the result follows. We observe that if u W, () then Du F = F( u) dx and it holds d dt u>t u F dx = P F ({u > t}; ). (43) Another important result that has been generalized to the anisotropic case is the isoperimetric inequality [4, 67, 68]. Proposition.4. If E is a set of finite perimeter in R n, then: P F (E; R n ) nκ n n E /n. (44) Proof. If E is a smooth set, then in [6] is proved the following P F (E; R n ) = Dχ E F = F(ν E n ) dσ nκ E /n, (45) R n E where ν E is the outer normal to E. follows. Then by Proposition.3 and (45), the result Now, we recall the useful definitions of anisotropic distance, diameter and inradius. We define the anisotropic distance function (or F-distance) to as d F (x) := inf y Fo (x y), x, and the anisotropic inradius as ρ F := max{d F (x), x }. We denote the diameter diam F of with respect to the norm F on R n as diam F () := sup F o (x y). (46) x,y It will be useful in the sequel an anisotropic version of the isodiametric inequality.

24 PRELIMINARIES Proposition.5. Let be a convex set in R n. Then κ n n diam F() n. (47) The equality sign holds if and only if is equivalent to a Wulff shape. Proof. We want prove that diam F () n n = diam F(W) n. κ n W We argue similarly as in [4, Th..]. Firstly, we observe that from definitions, it follows that has the same anisotropic diameter of its convex envelope, but it has a lower or equal volume. Hence, if we denote by C the convex envelope of, we have that diam F () n diam F( C ) n C. (48) Therefore, we can suppose that is a convex set and we prove that the minimum of the right hand side of (48) is reached by a Wulff shape. Let us suppose that diam F, we denote by the set that is symmetric to with respect to the origin and put B := ( + )/. The function t + ( t) /n, t, is concave so that = B and the equality sign holds only if is homothetic to, i.e. if has a center of symmetry. Let us call a and b the point that realize the diameter of B: F o (a b) = diam F B. Now, a = x + x /, b = y + y /, where x, y and x, y, hence: F o (a b) = Fo (x + x y y ) ( F o (x y) + F o (x y ) ) diam F + diam F and therefore diam F B. Now, it is sufficient to assume that has a center of symmetry. But then diam F () implies that is contained in Wulff shape of unit diameter, i.e. κ n / n. This in turn implies (47). Finally we observe that, in general, F and F o are not rotational invariant. Anyway, let us consider A SO(n) and define F A (x) = F(Ax). Since A T = A, then (F A ) o (ξ) = sup x R n \{} Moreover, we also have diam FA (A T ) = x ξ F A (x) = sup A T y ξ y R n \{} F(y) y Aξ = sup y R n \{} F(y) = (Fo ) A (ξ). sup (F o ) A (y x) = sup F o (ȳ x) = diam F (). x,y A T x,ȳ

25 A N I S OT R O P I C L A P L A C I A N E I G E N VA L U E P R O B L E M S In this chapther we analyze some properties of the eigenvalues of the anisotropic p-laplacian operator ( ) Q p u := div F p ( u) ξ F( u), (49) where F is a suitable smooth norm of R n and p ], + ]. We provide sharp estimates for eigenvalues of Q p u with both Dirichlet and Neumann boundary condition.. convex symmetrization for anisotropic elliptic equations with a lower order term.. Preliminary results In this Section, we estimate the solution of the eigenvalue problem of the anisotropic Laplacian with a lower order term, when Dirichlet boundary condition holds. We obtain comparison results with solutions of the convexly symmetric problem: { div(f( v) F( v)) b(f o (x))( F o (x) F( v))f( v) = f in v = in (5) where b is a suitable auxiliary function (see further for details). Firstly, we give three Lemmas, that are basic for our treatment. Lemma.. If u is any member of H (), then µ(t) n n κ n [ µ (t) ] [ d ] F ( u) (5) dt u >t for a.e. t such that < t < ess sup u. Proof. For h >, Schwarz inequality gives and F( u) ( ) ( ) dx F ( u) h t< u t+h h t< u t+h t< u t+h ( ) ( F( u) (µ(t) µ(t + h) F ( u)). h t< u t+h h h t< u t+h Therefore, as h +, we obtain d dt u >t F( u) ( µ (t) ) ( d F ( u)). dt u >t

26 ANISOTROPIC LAPLACIAN EIGENVALUE PROBLEMS By (44) and (43), we have ( nκn /n µ(t) n µ (t) d F ( u)). dt u >t Then squaring and dividing by n κ /n n µ(t) n, we obtain (5). Lemma.. E f E f (s) ds for any measurable set E. This Lemma is a special case of a theorem by Hardy and Littlewood (see [78], Theorem 378). Lemma.3. If ϕ is bounded and ϕ(t) + for a.e. t >, then ϕ(t) t + t K(s)ϕ(s) ds + ψ(t) ( s ) exp K(r) dr ( dψ(s)) t for a.e. t >. Here K is any nonnegative integrable function, ψ has bounded variation and vanishes at +. Lemma.3 is a generalization of Gronwall s lemma... Main result In this section we discuss our main result. It consists in showing that a solution to (5) can be compared in term of a solution to (5), where the function b is known as a pseudo rearrangement of B(x). It can be defined as b ( ( s κ n ) n ) = ( d B (x)), (5) ds u >u (s) We refer to [5] and [7] for further details. Theorem.4. Let u H () be a solution to the problem { div(a(x, u, u)) + b(x, u) = f in u = on (53) where a(x, η, ξ) {a i (x, η, ξ)} i=,...,n are Carathéodory functions satisfying a(x, η, ξ) ξ F (ξ) a.e. x, η R, ξ R n. (54) and b(x, ξ) is such that: b(x, ξ) B(x)F(ξ), (55)

27 CONVEX SYMMETRIZATION FOR ANISOTROPIC ELLIPTIC EQUATIONS 3 where B L k (), with k > n. We assume further that f L n n+ () if n 3; f L p (), p >, if n = ; F : R n [, [ is a convex function satisfying (7)-(8). Then u v (56) F q ( u) F q ( v) (57) with < q, and v(x) = ( ) /n κn F o (x) where b is defined as in (5). t t n dt ( r ) exp b(r )dr f (κ n r n )r n dr. (58) t Remark.5. The function in (58) is convexly symmetric, in the sense that v(x) = v (x). Indeed the function v (s) = s n κ /n n ( t ) ( t n κn t ) /n dt exp b(r )dr f (r)dr ( κn r ) /n is decresing and v(x) = v (κ n (F o (x)) n ). We observe that v(x) is a solution in H ( ) to the problem { div(f( v) F( v)) b(f o (x))( F o (x) F( v))f( v) = f in v = on. (59) In fact, if we define ρ = F o (x) and we look for a solution such that v(ρ) = v(f o (x)), we obtain v = v (ρ) ξ F o (x), (6) F( v) = v (ρ)f( ξ F o (x)) = v (ρ), (6) ξ F( v) = ξ F(v (ρ) ξ F o (x)) = ξ F( ξ F o (x)) = x F o (x). (6) A direct computation gives div(f( v) ξ F( v)) b(f o (x))f( v) ξ F o (x) ξ F( v) Using (58), we can write: v(ρ) = and we have: ( ) /n κn ρ t t n dt = v (ρ) n v (ρ) + ρ b(f o (x))v (ρ). ( t ) exp g(r )dr f (κ n r n )r n dr (63) ρ v (ρ) n v (ρ) + ρ b(f o (x))v (ρ) = f (ρ). (64) Collecting (63) and (64) we obtain that the function in (58) solves (59).

28 4 ANISOTROPIC LAPLACIAN EIGENVALUE PROBLEMS Remark.6. We can compute F q ( v). By (6) we have [F( v(x))] q = [v (ρ)] q = [ ρ ( ρ ) ] q ρ n exp b(r )dr f (κ n r n )r n dr r where ρ = F o (x). An integration by the substitution s = κ n r n gives [ ( [F( v(x))] q κn ρ n ) q ρ = nκ n ρ n exp b(r )dr f (s)ds], ( κn s ) /n therefore, by an integration on, we have [F( v(x))] q [ = nκ n ρ n κn ρ n exp Hence, by the sustitution τ = κ n ρ n, we have [F( v(x))] q = [ τ nκn /n τ n exp ( ρ ( s κn ) /n b(r )dr ( ( τ κn ) /n ) ( s κn ) /n b(r )dr ) f (s)ds] q dρ. f (s)ds] q dτ. Theorem.7. Let u H () be a solution to problem (53) under the assumption (54). Furthermore we suppose that (55) holds with B L () = β ; f L n n+ () if n 3; f L p (), p >, if n = ; F : R n [, [ is a convex function satisfying (7)-(8). Then (56) and (57) holds with v(x) = ( ) /n κn F o (x) t n dt t e β(r t) f (κ n r n )r n dr. (65) Remark.8. The function v(x) in (65) is a solution in H ( ) to the problem { div(f( v) ξ F( v)) βf( v) ξ F o (x) ξ F( v) = f in v = on. The proof of Theorem.7 is similar to that of Theorem.4 and it can be obtained from it considering the function B(x) as a constant...3 Proof of main Theorem Let us start by proving a preliminary result about the function b (see [7]). Lemma.9. If b is defined by (5), then ( d ) B (x) = µ dt (t) b u >t ( ( ) ) µ(t) n κ n (66)

29 CONVEX SYMMETRIZATION FOR ANISOTROPIC ELLIPTIC EQUATIONS 5 and d dt u >t B(x)F( u) d dt ( ) µ(t) n κn b(r)dr ( d ) F ( u) dt u >t (67) for almost every t [, ess sup u ]. Proof. Let p(t) and q(s) be the integrals of B(x) over { u > t} and { u > u (s)} respectively, hence p (t) = q (µ(t))µ (t) for almost every t [, ess sup u]. So equality (66) is proved. By Hölder inequality, we have d dt u >t by (66) we obtain d dt u >t B(x)F( u) B(x)F( u) ( d ) ( B(x) d F ( u)), dt u >t dt u >t µ (t) b ( ( µ(t) κ n ) ) n ( d F ( u)), dt u >t hence, by Lemma., d dt u >t B(x)F( u) µ (t) µ(t) n nκ /n n b ( ( µ(t) κ n ) ) n ( d ) F ( u), dt u >t that is equal to the right-hand side of (67). Proof of Theorem.4. Suppose u is a weak solution of problem (53), then a(x, u, u) ϕ + b(x, u)ϕ = f ϕ, ϕ H(). (68) For h >, t >, let ϕ be the following test function h, if u > t + h ϕ h (x) = u t, if t < u t + h, if u t, then, if u > t + h i ϕ h (x) = i u, if t < u t + h, if u t. Inserting this test function in (68), we have t< u t+h a(x, u, u) u + = u >t+h u >t+h b(x, u)h f h + ( f b(x, u))( u t) sign u. t< u t+h

30 6 ANISOTROPIC LAPLACIAN EIGENVALUE PROBLEMS The last term is smaller than ( f b(x, u))( u t) and, by hypothesis (54) t< u t+h and (55), we have F ( u) h B(x)F( u) f h t< u t+h u >t+h Dividing each term by h, as h +, (69) becomes d dt u >t and, by Lemma., d dt u >t Now, we can write B(x)F( u) = u >t F ( u) B(x)F( u) f, u >t u >t u >t+h + ( f b(x, u))( u t). t< u t+h (69) µ(t) F ( u) B(x)F( u) f (s)ds. (7) u >t + and hence, by Lemma.9, we have B(x)F( u) u >t + t d dt t ( ) µ(t) n κn Inserting (7) in (7) we obtain d dt + t u >t F ( u) d dt ( ) µ(t) n κn ( d ) B(x)F( u) ds ds u >s b(r)dr b(r)dr Now we can use Lemma.3 with ϕ(t) = d dt d + s F ( u) exp d dt dr u >t t ( d ) F ( u) ds. ds u >s ( d ) µ(t) F ( u) ds + f (s)ds. ds u >s t u >t F ( u). We have ( ) µ(r) n κn where ψ(s) = µ(s) f (ξ)dξ. Using the substitution ρ = µ(s) and σ = µ(r), we obtain ( ) µ(t) n d dt u >t F ( u) µ(t) exp κn ( σ κn ) n b(r )dr [ dψ(s)ds], (7) b(ρ)dρ f (σ)dσ. (7) Inequality (7) and Lemma. give µ(t) µ(t) n κ /n n ( µ (t)) exp n ( ) µ(t) n κn ( σ κn ) n b(ρ)dρ f (σ)dσ.

31 CONVEX SYMMETRIZATION FOR ANISOTROPIC ELLIPTIC EQUATIONS 7 for a.e. t [, ess sup u ], then integration of both sides with respect to t over the interval [, u (s)] yields u (s) s dt n κ /n n t ( t t n exp κn ) n ( σ κn ) n b(ρ)dρ f (σ)dσ (73) From formula (58), we learn that v (s) is the right-hand side of (73), so (56) is satisfied. In order to prove (57), we observe that Hölder inequality gives ( F q ( u) h t< u t+h h t< u t+h and hence, for t +, d dt u >t F q ( u) ( µ (t) ) q provided that < q. Lemma. gives [ d ] F ( u) dt u >t hence by inequality (7) nκ /n n ) q ( ) q dx F ( u) h t< u t+h ( d ) q F ( u), (74) dt u >t µ(t) n ( µ (t)) [ d ] F ( u), dt u >t [ d ] F ( u) dt u >t nκn /n µ(t) n ( µ (t)) µ(t) exp ( ) µ(t) n κn ( σ κn ) n b(ρ)dρ f (σ)dσ. (75) Coupling (75) with (74) d dt u >t F q ( u) ( µ (t)) q/ [ ( d dt u >t ( µ (t)) q/ µ(t) n ( µ (t)) nκ (/n) n ) ] q F ( u) µ(t) exp ( ) µ(t) n κn ( σ κn ) n b(ρ)dρ f (σ)dσ. q Consequently F q ( u) µ (t) µ(t) µ(t) n exp nκ (/n) n ( ) µ(t) n κn ( σ κn ) n b(ρ)dρ f (σ)dσ dt, q

32 8 ANISOTROPIC LAPLACIAN EIGENVALUE PROBLEMS and hence, by the substitution τ = µ(t), F q ( u) nκ (/n) n so the theorem is proved. τ ( τ τ n exp κn ) n ( σ κn ) n b(ρ)dρ f (σ)dσ dτ = F q ( v), q

33 ON THE SECOND ANISOTROPIC LAPLACIAN DIRICHLET EIGENVALUE 9. on the second dirichlet eigenvalue of some nonlinear anisotropic elliptic operators.. The Dirichlet eigenvalue problem for Q p In this Section, we study the second eigenvalue λ (p, ) of the anisotropic p-laplacian operator (49) with Dirichlet condition: { Qp u = λ(p, ) u p u in u = on. We provide a lower bound of λ (p, ) among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as p +. Firstly, we recall the following Definition.. A domain of R n is a connected open set. Here we state the eigenvalue problem for Q p. Let be a bounded open set in R n, n, < p < +, and consider the problem { Qp u = λ u p u in (76) u = on. Definition.. We say that u W,p (), u =, is an eigenfunction of (76), if F p ( u) ξ F( u) ϕ dx = λ u p uϕ dx (77) for all ϕ W,p (). The corresponding real number λ is called an eigenvalue of (76). Obviously, if u is an eigenfunction associated to λ, then F p ( u) dx λ = >. u p dx The first eigenvalue Among the eigenvalues of (76), the smallest one, denoted here by λ (p, ), has the following well-known variational characterization: F p ( ϕ) dx λ (p, ) = min. (78) ϕ W,p ()\{} ϕ p dx In the following theorems its main properties are recalled. Theorem.. If is a bounded open set in R n, n, there exists a function u C,α () C() which achieves the minimum in (78), and satisfies the problem (76) with λ = λ (p, ). Moreover, if is connected, then λ (p, ) is simple, that is the corresponding eigenfunctions are unique up to a multiplicative constant, and the first eigenfunctions have constant sign in. Proof. The proof can be immediately adapted from the case of connected and we refer the reader, for example, to [89, ].

34 ANISOTROPIC LAPLACIAN EIGENVALUE PROBLEMS Theorem.3. Let be a bounded open set in R n, n. Let u W,p () be an eigenfunction of (76) associated to an eigenvalue λ. If u does not change sign in, then there exists a connected component of such that λ = λ (p, ) and u is a first eigenfunction in. In particular, if is connected then λ = λ (p, ) and a constant sign eigenfunction is a first eigenfunction. Proof. If is connected, a proof can be found in [89, 47]. Otherwise, if u in disconnected, by the maximum principle u must be either positive or identically zero in each connected component of. Hence there exists a connected component such that u coincides in with a positive eigenfunction relative to λ. By the previous case, λ = λ (p, ) and the proof is completed. Here we list some other useful and interesting properties that can be proved in a similar way than the Euclidean case. Proposition.4. Let be a bounded open set in R n, n, the following properties hold.. For t > it holds λ (p, t) = t p λ (p, ).. If, then λ (p, ) λ (p, ). 3. For all < p < s < + we have p[λ (p, )] /p < s[λ (s, )] /s. Proof. The first two properties are immediate from (78). As regards the third property, the inequality derives from the Hölder inequality, similarly as in [9]. Indeed, taking φ = ψ s p ψ, ψ W,p () L (), ψ, we have by (7) that [λ (p, )] p s p ψ s p F p ( ψ)dx ψ s dx p s p By minimizing with respect to ψ, we get the thesis. In addition, the Faber-Krahn inequality for λ (p, ) holds. Theorem.5. Let be a bounded open set in R n, n, then F s ( ψ)dx ψ s dx p/n λ (p, ) κ p/n N λ (p, W). (79) Moreover, equality sign in (79) holds if is homothetic to the Wulff shape. The proof of this inequality, contained in [], is based on a symmetrization technique introduced in [4] (see [59, 64] for the equality cases). Using the previous result we can prove the following property of λ (p, ). Proposition.6. Let be a bounded domain in R n, n. The first eigenvalue of (76), λ (p, ), is isolated. Proof. We argue similarly as in [9]. For completeness we give the proof. For convenience we write λ instead of λ (p, ). Let λ k = λ a sequence of eigenvalues such that lim k + λ k = λ s

35 ON THE SECOND ANISOTROPIC LAPLACIAN DIRICHLET EIGENVALUE Let u k be a normalized eigenfunction associated to λ k that is, λ k = F p ( u k ) dx and u k p dx = (8) By (8), there exists a function u W,p () such that, up to a subsequence u k u in L p () u k u weakly in L p (). By the strong convergence of u k in L p () and, recalling that F is convex, by weak lower semicontinuity, it follows that u p dx = and F p ( u) dx lim k λ k = λ. Hence, u is a first eigenfunction. On the other hand, being u k not a first eigenfunction, by Theorem.3 it has to change sign. Hence, the sets + k = {u k > } and k = {u k < } are nonempty and, as a consequence of the Faber-Krahn inequality and of Theorem.3, it follows that λ k = λ (p, + k ) C n,f + k p n, λ k = λ (p, k ) C n,f k. p n This implies that both + k and k cannot vanish as k + and finally, that u k converges to a function u which changes sign in. This is in contradiction with the characterization of the first eigenfunctions, and the proof is completed. Higher eigenvalues First of all, we recall the following result (see [69, Theorem.4.] and the references therein), which assures the existence of infinite eigenvalues of Q p. We use the following notation. Let S n be the unit Euclidean sphere in R n, and M = {u W,p () : u p dx = }. (8) Moreover, let C n be the class of all odd and continuous mappings from S n to M. Then, for any fixed f C n, we have f : ω S n f ω M. Proposition.7. Let be a bounded open set of R n, for any k N, the value λ k (p, ) = inf f C n max ω S n is an eigenvalue of Q p. Moreover, and F p ( f ω )dx < λ (p, ) = λ (p, ) λ (p, )... λ k (p, ) λ k+ (p, )..., λ k (p, ) as k. Hence, we have at least a sequence of eigenvalues of Q p. following proposition holds. Furthermore, the Proposition.8. Let be a bounded open set of R n. The spectrum of Q p is a closed set.